First slide

Algebra of vectors

Question

If 'I' is the incentre of $∆$ ABC whose corresponding sides are a, b, c, respectively, then $\vec{IA} + b \vec{IB} + c \vec{IC}$ is always equal to

Moderate

A

$\vec{0}$
B

$(a + b + c) \vec{BC}$
C

$(\vec{a} + \vec{b} + \vec{c}) \vec{AC}$
D

$(a + b + c) \vec{AB}$

Solution

Let the incentre be at the origin and be $A (\vec{p}), B (\vec{q}) and C (\vec{r})$ then $\vec{IA} = \vec{p}, \vec{IB} = \vec{q} and \vec{IC} = \vec{r}$
Incentre I is $\frac{a \vec{p} + b \vec{q} + c \vec{r}}{a + b + c}$ where p = BC, q=AC and r=AB
Incentre is at the origin.
Therefore,
$\begin{array}{l} \frac{a \vec{p} + b \vec{q} + c \vec{r}}{a + b + c} = \vec{0}, or a \vec{p} + b \vec{q} + c \vec{r} = \vec{0} \\ \Rightarrow a \vec{IA} + b \vec{IB} + c \vec{IC} = \vec{0} \end{array}$

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